reduction graph
The Riemann Hypothesis
Derived, and unstated
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Grand Riemann Hypothesis implies The Riemann Hypothesis
What would settle it
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Grand Riemann Hypothesis implies the target
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Generalized Riemann Hypothesis implies the target
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A Hilbert-Polya spectral realization implies the target
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The Mertens Conjecture implies the target
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conversely, the mere existence of any moderate net of this damped-Báez–Duarte form that is uniformly… implies the target
conversely, the mere existence of any moderate net of this damped-Báez–Duarte form that is uniformly L²-bounded and associated with −χ_{(0,1)} forces the Riemann hypothesis.
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A finite spectral averaging framework replaces this obstruction by proving cancellation at the critical… implies the target
A finite spectral averaging framework replaces this obstruction by proving cancellation at the critical line, positivity of the off-critical paired contribution on the left under a concrete low-frequency kernel condition, a conditional zero-density consequence, and a precise statement of the additional localization hypotheses needed to imply the Riemann Hypothesis.
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the two standpoints are offered as complementary routes toward clarifying the operator’s still-unsettled… is partial progress toward the target
the two standpoints are offered as complementary routes toward clarifying the operator’s still-unsettled connection to the Riemann hypothesis.
What it would settle
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the target is equivalent to von Koch prime counting error term
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the target is equivalent to Square-root cancellation in the Mertens function
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the target is equivalent to Robin's inequality
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the target is equivalent to Lagarias' criterion
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the target is equivalent to The Nyman-Beurling criterion
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the target is equivalent to Li's criterion
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the target is equivalent to The de Bruijn-Newman constant
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the target is equivalent to The Franel-Landau criterion
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the target is equivalent to Weil positivity
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the target implies The Lindelof Hypothesis
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the target implies The quasi-Riemann hypothesis
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the target implies The Density Hypothesis
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the target is equivalent to The paper shows that for certain pairs of normalized error terms, such as those associated with ψ(x) and the…
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the target is equivalent to The Riemann hypothesis is equivalent to the validity of a collection of estimates involving an operator…
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the target is equivalent to The main result is that the property 'For all n > 0, log g(n) < li^{-1}(n)' (where g(n) is the maximal order…
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the target is equivalent to For any fixed c>0, the Riemann hypothesis is equivalent to the assertion that the local moments M_{q,c}(N)…
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the target is equivalent to Under the Deep Riemann Hypothesis, for any fixed spectral scale T the difference between the mollified…
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the target is equivalent to RH holds if and only if the trajectory error functional satisfies E(X) ≪ X^(1/2) log X for all X ≥ e^120.
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the target is equivalent to if that uniqueness holds, it characterizes the zeta zeros intrinsically without settling the Riemann…
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the target is equivalent to dRe xi/dt, dIm xi/dt) equals the sign of epsilon for every point s=1/2+epsilon+it in the critical strip if…
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the target is equivalent to the Riemann hypothesis is true if and only if a_{1/2}(n)=sigma_{1/2}(n)/(n^{1/2} exp(li((log n)^{1/2}))) is…
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the target is equivalent to for i=1,2, the Riemann hypothesis holds if and only if the integral from 2 to X of the ith Mertens error…
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the target implies Modifying the work of Klagsbrun, Mazur, and Rubin, the authors prove that under the extended Riemann…
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the target implies we show that for k>0, (1/T) meas{t in [T,2T]:
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the target implies we prove the result holds for every n > 24.
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the target implies the poles activate at times t_n ∝ ρ_n³ and condense into an infinite-time essential singularity.
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the target implies As an application, we break the persistent practical barrier around 0.515 and prove μ < 0.50895 under the…
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the target implies Σ_p ||M_p(n)||²_{L∞} ≤ C M_n² n^{4/3+o(1)}.
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the target implies the authors calculate that the logarithmic density of the set where two specific error terms have the same…
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the target implies for large n, B(n) = O(n^{3/4} (log n)^{1/2}) and L(n) ≥ c' n^{1/4} (log n)^{-1/2}.
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the target implies the normalized correlation ⟨μ(n)M(n-1)⟩(T) equals −3/π²(1 − T^{(c−1)δ(T)}) plus the sum from 0 < γ < T of 1…
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the target implies χ) and that the non-trivial zeros ρ=½+iγ of L(s,χ) are simple, the discrete moments ∑_{0<γ≤T} |L'(ρ,χ)|^{-2}…
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the target implies the two damped nets are moderate, uniformly L²-bounded, and associated with −χ_{(0,1)};
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the target implies the sum over those zeros of the squared modulus of a fixed ratio of zeta functions is at least half the…
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the target implies Schrödinger operators with anisotropic potentials, Dirichlet Laplacians on infinite-volume domain
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the target implies in the abelian case and under the Riemann Hypothesis for Dirichlet L-functions a more precise error estimate…
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the target implies Assuming the Riemann Hypothesis and a mild lower bound on the Gaussian smoothing width relative to a fixed…
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the target implies For 1<α_1<...<α_r, the vector (log ζ(s+i(log τ)^{α_1}),...,log ζ(s+i(log τ)^{α_r})) is jointly universal in…
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the target implies for all t ≥ e^18.
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the target implies for every 0<δ<1 and x≥e^e, |E^AN_{σ1}(x)| ≪ x^{δ'} exp(log x / log log x) with δ'=max{1/2,δ}.
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the target implies SZC, and a convergence condition) that A_f^exp(x) - Δ_1(x) = Δ_{1/2}(x) + Σ_ρ Δ_ρ(x) + O(x^a), where Δ_1,…
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the target implies for any fixed spectral scale T the difference between the mollified prime-power sums for 1 and a (mod N),…
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the target implies Conjecture 1 states that the discrete shifted moment Σ_{0<γ≤T} ζ(1/2+iγ+α_1)···ζ(1/2+iγ+α_k) equals the…
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the target implies for Re(k)>−3, the average of ζ′(1/2+iγ)^k over zeros γ with 0<γ≤T is asymptotically (log(T/2π))^k / Γ(k+2).
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the target implies define $P_{k/2}(T)$ as the normalized count of pairs of zeros with imaginary parts in $[T/\log^2 T, T]$…
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the target implies f)$ and Dirichlet $L$-functions, the long-interval proportion rises to at least $1/2$, and for $1/2<\mu<1$…
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the target implies Theorem 1.2 achieves the same factorization for the imaginary part with the full support $\eta<2/k$;
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the target implies so the lower bound is unconditional.
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the target implies $\mu(\theta)\le 1-\theta/2$ under the Lindelöf hypothesis, and $\mu(\theta)\le 1-\theta/12$ under the…
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the target implies all zeros of $\zeta(2s)$ lie on $\Re(s)=1/4$ and are simple, so the pole sum runs over that line up to a…
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the target implies the pair correlation conjecture (Conjecture 1.2), the triple correlation conjecture (Conjecture 1.3), and…
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the target implies N_r(T,c), the number of ordinates γ_n ≤ T whose next r gaps are all at least 2πc/log T, satisfies N_r(T,c) ≥…
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the target implies For the third Mertens error term, RH implies the same positivity, and if the supremum of the real parts of…
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the target implies and $\mu(\theta)\le 1-\theta/12$ under the density hypothesis.
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the target implies The same framework recovers the classical thresholds for the prime number theorem in all short intervals and…
Named statements in this graph
- Generalized Riemann Hypothesis
- The Density Hypothesis
- The Lindelof Hypothesis
- Grand Riemann Hypothesis
- The quasi-Riemann hypothesis
- A Hilbert-Polya spectral realization
- Lagarias' criterion
- Li's criterion
- Montgomery's pair correlation conjecture
- Robin's inequality
- Square-root cancellation in the Mertens function
- The de Bruijn-Newman constant
- The Franel-Landau criterion
- The Mertens Conjecture refuted
- The Nyman-Beurling criterion
- von Koch prime counting error term
- Weil positivity